AUDA

Hi Russel,

On 15 Feb 2009, at 03:41, russell standish wrote:


By Godel's theorems, Löb's theorems and Solovay theorems.

When B is seen as provability, it really means "I can prove that p is  
true"  *when* asserted by a self referentially correct universal  
machine, believing (or asserting, or proving) the induction axioms,  
like Peano Arithmetic, or Zermelo-Fraenkel Set Theory.

In that case the "B" is the corresponding (to PA, ZF, any Lobian  
Machine) Gödel provability predicate, and it obeys invariant  
mathematical provability laws.
It can be proved that the modal logic G is sound and complete for the  
arithmetical provability laws that the machine can prove, and it can  
be proved that the modal logic  G* is sound and complete for the true  
provability laws, with the propositional variable "p" interprete by  
closed arithmetical formula. I will (re)come back on this later  
probably.

The soundness is a quasi direct consequence of Gödel's and Löb's  
incompleteness theorems, together with the fact that the modal rule of  
inference preserve the arithmetical modal provability and correctness  
(for G and G* respectively).

The completeness is far more complex to prove, and it has been done by  
Solovay theorem. (Reference in my Post to Günther, or in my both  
thesis where this is explained with various details. The proof of  
Solovay is explained in Smorynski book, and in Boolos 1979 and 1993  
books in details. It is the base of AUDA, or the "interview of the  
Lobian Machine" (see any of my papers or theses).


You get the hypostases by introducing the Theaetical intensional  
nuances (Bp & p, Bp & Dp, Bp & Dp & p, etc.). G is really the third  
personne self-reference, "Bp & p" gives the first person self-
reference: the real one that the machine cannot even name. Bp & Dp  
gives intelligible matter, Bp & Dp & p gives the sensible matter, Etc.  
G* knows them equivalent, proving the same arithmetical theorem, but  
they obeys veruy different logics due to the fact that the machine  
cannot know them equivalent.
And you get the computationalist hypostases by restricting the  
interpretation of the propositional variables, p, to the Sigma-1  
sentences (which proofs provide the arithmetical Universal Dovetailing).

All this is the substance of AUDA, which leads to the theology  
(including the verifiable physics or the logic of the observable) of  
the Universal machine.
The corona G* minus G and its intensional variants, give the proper  
theological reality of the universal machine. What they can correctly  
hope or fear, and correctly bet, without being to prove.

See the second part of the Sane2004 paper for a short but precise  
account of AUDA (the first part is UDA(1...8).
See the Plotinus paper for a longer explanation of AUDA, and to see  
its use for providing an arithmetical interpretation of both Plotinus  
"theory of mind" and "theory of matter", or see the two thesis for  
detailed explanation and motivations. Or my old post on "the machine-
itself" and its "Guardian Angel" (if you remind?). Or wait to see if  
i will succeed to explain this on the list,  or out of the list. I  
have already tried with not so much success, but those matter are not  
so easy of course. But an understanding of UDA + an understanding of  
those mathematical theorems leads "automatically" to AUDA. And to the  
discovery that, about machine only,  there is already no unifying  
complete everything theory. On the contrary you will see that all  
universal machine which introspects herself discover or create an  
universal Everything *realm* in its own "head" ...

Best,

Bruno



http://iridia.ulb.ac.be/~marchal/




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